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Dynamics of Complex Unicritical Polynomials

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Dynamics of Complex Unicritical Polynomials Dynamics of complex unicritical polynomials A Dissertation Presented by Davoud Cheraghi to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics Stony Brook University August 2009 Stony Brook University The Graduate School Davoud Cheraghi We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation. Mikhail M. Lyubich Professor of Mathematics, Stony Brook University Dissertation Advisor John Milnor Distinguished Professor of Mathematics, Stony Brook University Chairperson of Defense Dennis Sullivan Distinguished Professor of Mathematics, Stony Brook University Inside Member Saeed Zakeri Assitant Professor of Mathematics, City University of New York Outside Member This dissertation is accepted by the Graduate School. Lawrence Martin Dean of the Graduate School ii Abstract of the Dissertation Dynamics of complex unicritical polynomials by Davoud Cheraghi Doctor of Philosophy in Mathematics Stony Brook University 2009 In this dissertation we study two relatively independent problems in one dimensional complex dynamics. One on the parameter space of unicritical polynomials and the other one on measurable dynamics of certain quadratic polynomials with positive area Julia sets. It has been conjectured that combinatorially equivalent non-hyperbolic uni- critical polynomials are conformally conjugate. The conjecture has been al- ready established for finitely renormalizable unicritical polynomials. In the first part of this work we prove that a type of compactness, called a priori bounds, on the renormalization levels of a unicritical polynomial, under a cer- tain combinatorial condition, implies this conjecture. It has been recently shown that there are quadratic polynomials with pos- itive area Julia set. The second part of this work investigates typical trajecto- ries of quadratic polynomials with a non-linearizable fixed point of high return iii type. This class contains the examples with positive area Julia set mentioned above. In particular, we prove that Lebesgue almost every point in the Julia set of these maps accumulates on the non-linearizable fixed point. We show that the post-critical sets of these maps, which are the measure theoretic at- tractors, have measure zero and are connected subsets of the plane. This has some interesting corollaries such as, almost every point in the Julia set of such maps is non-recurrent, or, there is no finite absolutely continuous invariant measure supported on the Julia set of these maps. iv Contents ListofFigures............................... vii Acknowledgements ............................ viii 1 Introduction 1 2 Preliminaries 11 2.1 JuliaandFatousets ........................ 11 2.2 Dynamics of polynomials . 14 2.3 Unicritical family and the connectedness locus . 15 2.4 Quasi-conformalmappings . 20 2.5 Polynomial-like maps . 22 2.6 Holomorphicmotions........................ 24 3 Combinatorial Rigidity in the Unicritical Family 25 3.1 Modifiedprincipalnest . .. .. 25 3.1.1 Yoccozpuzzlepieces . .. .. 25 3.1.2 The complex bounds in the favorite nest and renormal- ization............................ 26 3.1.3 Combinatoricsofamap . 32 3.2 Proofoftherigiditytheorem. 35 3.2.1 Reductions ......................... 35 v 3.2.2 Thurstonequivalence . 37 3.2.3 The domains Ωn,j and the maps hn,j ........... 41 i 3.2.4 The gluing maps gn(k) ................... 58 3.2.5 Isotopy............................ 62 3.2.6 Promotiontohybridconjugacy . 67 3.2.7 Dynamical description of the combinatorics . 69 4 Typical Trajectories of complex quadratic polynomials 72 4.1 Accumulationonthefixedpoint. 72 4.1.1 Post-criticalsetasanattractor . 72 4.1.2 The Inou-Shishikura class and the near-parabolic renor- malization.......................... 75 4.1.3 Sectorsaroundthepost-criticalset . 80 4.1.4 Sizeofthesectors...................... 89 4.1.5 PerturbedFatoucoordinate . 93 4.1.6 Proof of main technical lemmas . 106 4.2 Measureandtopologyoftheattractor . 111 4.2.1 Balls in the complement . 111 4.2.2 Going down the renormalization tower . 116 4.2.3 Going up the renormalization tower . 121 4.2.4 Proofofmainresults . .132 Bibliography ..............................135 vi List of Figures 2.1 TheMandelbrotset......................... 17 2.2 The connectedness locus and a primary limb in it . 18 M3 2.3 A primitive renormalizable Julia set and the corresponding little multibrotcopy ........................... 19 3.1 The multiply connected domains and the buffers . 41 3.2 Primitivecase............................ 43 3.3 An infinitely renormalizable Julia set . 50 3.4 A twice satellite renormalizable Julia set . 53 4.1 A schematic presentation of the Polynomial P .......... 76 4.2 An example of a perturbed Fatou coordinate and its domain. 78 4.3 Thefirstgenerationofsectors . 82 4.4 The domain ΣC2 ..........................100 vii Acknowledgements First of all, I would like to thank my adviser Mikhail Lyubich for his great patience and support during my graduate studies. He introduced many inter- esting subjects in holomorphic dynamics to me and was particularly helpful with improving my mathematics writing. Especially, his clear lectures have al- ways been a source of enthusiasm for me and for many others in the dynamics community at the Stony Brook university. I would like to thank the people at the Mathematics Department and IMS at Stony Brook for their generous support. I have enjoyed talking to and learning from many friends there. In particular, I want to thank M. Ander- son, C. Bishop, C. Cabrera, H. Hakobian, M. Martens, J. Milnor, S. Sullivan, S. Sutherland, S. Zakeri. R. Roeder was particularly helpful with improving my mathematics and English writing. I had privilege of spending a year of my studies at the Fields Institute in Toronto. I would like to thank them for their support. Specially, many lengthy conversations with C. Chamanara, J. Kahn, R. Perez, D. Schleicher, M. Shishikura, M. Yampolsky at the Institute created excellent memories dur- ing that year. Many thanks go to my great family to whom I owe every thing I have accomplished. Especially, to my parents for their trust in me and the freedom they gave me during all my life, to my brother and sisters for their constant encouragement and the confidence they gave me. Chapter 1 Introduction In this work we study two different problems in one dimensional holomorphic dynamics. The first one is on combinatorial rigidity of a class of unicritical maps. The second one describes typical (with respect to the Lebesgue measure) trajectories of certain quadratic polynomials with positive area Julia set. Below we describe the content of these two studies. Part 1. Dynamics of a rational map on the Riemann sphere breaks into two pieces; dynamics on the Fatou set and dynamics on the Julia set. Fatou set of a map is “nicely behaved” part of the dynamics, and it is completely understood through works of Fatou [Fat19], [Fat20a], [Fat20b], Julia [Jul18] and Sullivan [Su85]. The Julia set of a rational map is the “chaotic” part of the dynamics. Substantial work by many people has been devoted to under- standing these two sets for different families of rational maps. Basic properties of these sets has been collected in several books and surveys: See for example Beardon [Be91], Milnor [M06], Blanchard [Bl84], and Lyubich [Ly86]. Hyperbolic maps form a class of thoroughly understood and well behaved maps. These are rational maps for which the orbits of all critical points tend to attracting periodic points. For every map in this class, there exists a finite 1 subset of the Riemann sphere such that a full measure set of points on the sphere is attracted to this set under dynamics of the map. One of the main conjectures in holomorphic dynamics, which goes back to Fatou, states that the hyperbolic maps are dense in the space of rational maps of degree d (also in the space of polynomials of degree d). By an approach developed through the work of R. M˜ane, P. Sad and D. Sullivan [MSS83] for rational maps to tackle this problem, “studying families of rational maps has been reduced to studying of dynamics of individual maps on their Julia set”. Every polynomial with one critical point, which is referred as unicritical polynomial through this thesis, can be written (up to conformal conjugacy) in the form z zd + c, for some complex number c. If the critical point → 0 tends to infinity under iteration of such a map (attracted to the super- attracting fixed point at infinity), the map is hyperbolic. The Julia set is a Cantor set in this case. Therefore, attention reduces to parameters for which orbit of 0 stays bounded under iteration. The set of such parameters is called the connectedness locus or the Mulribrot set, (the well-known Mandelbrot set corresponds to d = 2) see Figures 2.1 and 2.2. There is a way of defining graded partitions of the Multibrot set into pieces such that dynamics of maps in each piece have some special combinatorial property. All maps in a given piece of partition of a certain level will be called combinatorially equivalent up to that level. Conjecturally, combinatorially equivalent (up to all levels) non-hyperbolic maps in the uniciritcal family are conformally conjugate. As stated in [DH82] for d = 2, this Rigidity Conjecture is equivalent to the local connectivity of the Mandelbrot set (MLC for short) and naturally extends to degree d unicritical polynomials. In [DH82], they prove that MLC implies Density of hyperbolic quadratic polynomials
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